A273449 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 726", based on the 5-celled von Neumann neighborhood.

1, 5, 9, 21, 21, 53, 53, 73, 85, 141, 141, 161, 185, 257, 261, 229, 297, 445, 421, 437, 497, 597, 597, 557, 657, 965, 925, 961, 1029, 1217, 1237, 1189, 1349, 1665, 1741, 1729, 1829, 1849, 2025, 1957, 2117, 2569, 2637, 2585, 2829, 2769, 2993, 2861, 3149, 3505

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Robert Price, Table of n, a(n) for n = 0..128

Robert Price, Diagrams of the first 20 stages

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=726; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Crossrefs

Sequence in context: A272786 A273150 A273388 * A297362 A175364 A244642

Adjacent sequences: A273446 A273447 A273448 * A273450 A273451 A273452

Keyword

nonn,easy

Author

Robert Price, May 22 2016

Status

approved